The ordinary Grassmannian of k-planes in n-space is a coset space for $GL_n$. It is $GL_n$ mod a maximal parabolic. Here there is a nice basis given by Schubert varieties, which can be indexed by Young diagrams that fit in an (k)x(n-k) box. The structure constants for the cup product are then given by Littlewood-Richardson numbers.
My question: is there a similarly nice picture for Grassmannians of arbitrary simple groups. Here the ordinary Grassmannian is replaced by $G/P$ where $G$ is a simple group and $P$ is a maximal parabolic. There are still Schubert varieties in this case, but I don't know how to say anything about the cup product.